We will use the notes by Professor Eric Carlen in this course - - - we will provide the electronic version soon. The syllabus and the notes of this course are very ambitious and intensive ! Some may assume that this course is an honors version of 251, but this course runs on a much more demanding level than a typical honors version of a regular course - - it covers at least 30% more material than a regular version and have a more conceptual component.
Professor Carlen's notes are fairly different from a traditional multivariable text, and demand careful readings and working through. If you find certain material difficult to comprehend from Professor Carlen's notes, you may consult some traditional texts, such as the one used 251, Calculus, Early Transcendentals, 3rd edition by Jon Rogawski & Colin Adams. Another useful source, somewhat closer to Professor Carlen's approach, is Vector Calculus, Linear Algebra, And Differential Forms, A Unified Approach by John H. Hubbard and Barbara B. Hubbard (There are five editions now; you may consult relevant sections of chapters 1-3 of any edition for the topics in our course. A similar comment applies to the text by Jon Rogawski & Colin Adams---the first and second edition has Jon Rogawski as the sole author).
In a traditional multivariable course such as 251 only the calculus of functions of two or three variables are discussed; linear algebra or even matrix operations are usually not introduced. 291 aims to discuss the calculus of vector-valued functions of n variables for an arbitrary n, so using matrix operations and other linear algebra operations would make it more clear. Linear algebra also plays such a prominent role in many other scientific, engineering, or even social science fields (such as in data representation and analysis). For this reason some essential part of linear algebra is introduced in this course in an integral way and is applied in studying the geometry of multi-dimensional Euclidean space and the calculus of vector-valued functions of multivariables.
The text of Jon Rogawski & Colin Adams does a good job discussing the geometry of two or three dimensional Euclidean spaces and the associated calculus---your won't find discussions on the linear algebra here. Professor Carlen's notes and the text by John H. Hubbard and Barbara B. Hubbard provide a more conceptual discussion of the underlying mathematics, in addition to covering the calculus of vector-valued functions of n variables for an arbitrary n. John H. Hubbard and Barbara B. Hubbard used a metaphor that learning multi-variable calculus in a traditional course is like learning how to drive a car, while they aim to teach how a car is built and why it works the way it does, in addition to teaching how to drive a car. This comment applies to this course.
Note: We will use every other Th6 recitation session (on average) for lecture in 291, so we expect to have about 33 lectures in total, instead of the typical 26 lectures. *ed sections denote more conceptual sections of linear algebra which deserve additional study outside of this course.
| Syllabus for 640:291 | |||
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| Lecture | Workshops/Challenge Problem Sets/Recitations | 291 Topic(s) and text sections | |
| 1 | 1.1.1 Geometry, Algebra and Calculus 1.1.2 Vector variables and Cartesian coordinates 1.1.3 Parameterization 1.1.4 The vector space R^n (September 4) |
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| 2 | 1.1.5 Geometry and the dot product 1.1.6 Parallel and orthogonal components 1.1.7 Orthonormal subsets of R^n (September 5) |
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| 3 |
1.1.8* Householder reflections and orthonormal bases 1.2.1 The cross product in R^3 1.2.2 Lines and planes in R^3 1.2.3 Distance problems (September 9) |
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| 4 | First Challenge Problem workshop (September 12) |
1.3.1 The Gram-Schmidt Orthonormalization Algorithm in R^3 1.3.2 The Gram-Schmidt Algorithm in general 1.3.3* Subspaces of R^n 1.3.4* Orthogonal complements (September 11) |
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| 5 | 2.1.1 Continuity of functions from R to R^n 2.1.2 Differentiability of functions from R to R^n 2.1.3 Velocity and acceleration (September 16) |
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| 6 | 2.1.4 Torsion and the Frenet-Seret formulae for a curve in R^3
2.1.5 Curvature and torsion are independent of parameterization 2.1.6 Speed and arclength 2.1.7 Speed, curvaure and torsion are independent of the choice of a right-handed coordinate system (September 18) |
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| 7 | 3.1.1 Functions of several variables 3.1.2 Continuity in several variables 3.1.3 Continuity and limits (September 19) |
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| 8 | 3.1.4 The Squeeze Principle in several variables 3.1.5 Continuity versus separate continuity (September 23) |
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| 9 | Second Challenge Problem workshop (September 26) | 4.1.1 Directional derivatives and partial derivatives 4.1.2 The gradient and a chain rule for functions of a vector variable 4.1.3 The geometric meaning of the gradient 4.1.4 Critical points 4.1.5 The gradient and tangent planes (September 25) |
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| 10 |
4.2.1 The matrix representation of linear functions 4.2.2 Composition of linear functions and matrix multiplication (September 30) |
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| 11 | 4.3.1 Differentiability and best linear approximation in several variables 4.3.2 The general chain rule (October 2) |
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| 12 | Review (To be scheduled) |
5.1.1 Implicit and explicit descriptions of planar curves 5.1.2 When is the contour curve actually a curve? 5.2.1 Lagrange's criterion for optmizers on the boundary (October 3) |
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| 13 |
5.2.2 Application of Lagrange's Theorem 6.1.1 The matrix form of a purely quadratic function (October 7) |
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| 14 | Exam 1 (Oct. 9, timing approximate!) | ||
| 15 | 6.2.1 Higher order directional derivatives and repeated partial differentiation 6.2.2 Clairault's Theorem 6.2.3 A multi-variable second order Taylor expansion (October 10) |
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| 16 | 6.1.2 Purely quadratic functions as sums of squares 6.1.3 Eigenvalues and eigenvectors of a symmetric matrix (October 14) |
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| 17 | Recitation (October 17) |
4.2.3 Solving the equation Ax=b 4.2.4 QR factorization (October 16) |
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| 18 | 4.2.5 Matrix inverses 6.1.4* Computing eigenvectors and eigenvalues (October 21) |
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| 19 | Third Challenge Problem workshop (October 24) |
6.2.5 Contour plots near critical points 6.2.6 Types of critical points for real valued functions on R^n (October 23) |
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| 20 | 7.1.1 A look back at integration in one variable 7.1.2 Integrals of functions on R^2 7.1.3 Computing area integrals (October 28) |
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| 21 | 7.1.4 Polar coordinates 7.2.1 Letting the boundary of D determine the disintegration strategy (October 30) |
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| 22 | 7.2.2 The change of variables formula for integrals in R^2 (October 31) | ||
| 23 | 7.3.1 Reduction to iterated integrals in lower dimension 7.3.2 The change of variables formula for integrals in R^3 (November 4) |
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| 24 | Fourth Challenge Problem workshop (November 7) | 7.4.1 Parameterized surfaces 7.4.2 The surface area of a parameterized surface (November 6) |
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| 25 | 9.1 Flows and flux 9.2 The Divergence Theorem (November 11) |
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| 26-27 | 9.2 The Divergence Theorem (continued) (November 13; Review on November 14) | ||
| 28 | Exam 2 (November 18, timing approximate!) | ||
| 29 | Recitation (November 21) |
9.3 Line integrals, force fields and work 9.3.1 Conservative vector fields (November 20) |
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| 30-31 | 9.3.2 Curl, circulation and Stokes' Theorem (November 25, 26--Change in Designation of Class) | ||
| 32 | 9.3.3 Curl and conservative vector fields 9.3.4 Vector potentials 9.4 The Laplace operator and Poisson's Equation (December 2) |
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| 33 | Recitation (December 5) | 4.4 Newton's Method (December 4) | |
| 34-35 | 5.3.1 Inverting coordinate tranformations 5.3.2 From the Inverse Function Theorem to the Implicit Function Theorem 5.4 The general Implicit Function Theorem (December 9; Review on December 11; Final Exam, noon--3pm, December 17) |
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